How to Integrate Complex Vectors Using the Vector Triple-Cross-Product Formula?

[Ceres629]

$$\int [a(\dot{b}\cdot a + b\cdot\dot{a}) + \dot{a}(b\cdot a) - 2(\dot{a}\cdot a)b - \dot{b}|a|^2]\, dt$$

The above are all vectors. How would one go about integrating this, the answer is apparently

$$a \times (a \times b) + h$$

where h is a constant vector

I don't quite see how they arrive at this answer...

 

[tiny-tim]

p x (q x r) = q(p.r) - r(p.q)

$$a \times (a \times b) + h$$

Hi Ceres629!

Easy-peasy … use the standard vector triple-cross-product formula:

p x (q x r) = q(p.r) - r(p.q), and then differentiate carefully!